I am trying to expose the joists in my living but don't want the wood look in between them. My home has an apartment upstairs so I need to sound proof the best that I can. I've considered using extruded polystyrene and then drywall over that...I'm hoping that someone might have some other suggestions because this ends up being fairly thick, and takes away from the exposed joists.
A lot of people paint their floor joists dark green, black or dark red, which makes them look pretty cool.
There isn't much that you'll be able to do to soundproof the structure without sheetrock on the underside. What you propose will do little or nothing to reduce sound transmission because of the amount of rigid contact between the materials. Sound attenuation batts combined with sheetrock (mounted under the floor joists) attached using resilient channel would be the most effective.